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Theorem List for Metamath Proof Explorer - 2401-2500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsbbid 2401 Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))

Theoremsblbis 2402 Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥](𝜒𝜑) ↔ ([𝑦 / 𝑥]𝜒𝜓))

Theoremsbrbis 2403 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥](𝜑𝜒) ↔ (𝜓 ↔ [𝑦 / 𝑥]𝜒))

Theoremsbrbif 2404 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑥𝜒    &   ([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥](𝜑𝜒) ↔ (𝜓𝜒))

Theoremsbequ8ALT 2405 Alternate proof of sbequ8 1883, shorter but requiring more axioms. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦𝜑))

Theoremsbie 2406 Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 13-Jul-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)

Theoremsbied 2407 Conversion of implicit substitution to explicit substitution (deduction version of sbie 2406). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))

Theoremsbiedv 2408* Conversion of implicit substitution to explicit substitution (deduction version of sbie 2406). (Contributed by NM, 7-Jan-2017.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))

Theoremsbcom3 2409 Substituting 𝑦 for 𝑥 and then 𝑧 for 𝑦 is equivalent to substituting 𝑧 for both 𝑥 and 𝑦. (Contributed by Giovanni Mascellani, 8-Apr-2018.) Remove dependency on ax-11 2032. (Revised by Wolf Lammen, 16-Sep-2018.) (Proof shortened by Wolf Lammen, 16-Sep-2018.)
([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)

Theoremsbco 2410 A composition law for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.)
([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremsbid2 2411 An identity law for substitution. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑥𝜑       ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)

Theoremsbidm 2412 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jul-2019.)
([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremsbco2 2413 A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.)
𝑧𝜑       ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremsbco2d 2414 A composition law for substitution. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑 → Ⅎ𝑧𝜓)       (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))

Theoremsbco3 2415 A composition law for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 18-Sep-2018.)
([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Theoremsbcom 2416 A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 20-Sep-2018.)
([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑)

Theoremsbt 2417 A substitution into a theorem yields a theorem. (See chvar 2260 and chvarv 2261 for versions using implicit substitution.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Jul-2018.)
𝜑       [𝑦 / 𝑥]𝜑

Theoremsbtrt 2418 Partially closed form of sbtr 2419. (Contributed by BJ, 4-Jun-2019.)
𝑦𝜑       (∀𝑦[𝑦 / 𝑥]𝜑𝜑)

Theoremsbtr 2419 A partial converse to sbt 2417. If the substitution of a variable for a non-free one in a wff gives a theorem, then the original wff is a theorem. (Contributed by BJ, 15-Sep-2018.)
𝑦𝜑    &   [𝑦 / 𝑥]𝜑       𝜑

Theoremsb5rf 2420 Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 20-Sep-2018.)
𝑦𝜑       (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))

Theoremsb6rf 2421 Reversed substitution. (Contributed by NM, 1-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 21-Sep-2018.)
𝑦𝜑       (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))

Theoremsb8 2422 Substitution of variable in universal quantifier. (Contributed by NM, 16-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
𝑦𝜑       (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Theoremsb8e 2423 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
𝑦𝜑       (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Theoremsb9 2424 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) Allow a shortening of sb9i 2425. (Revised by Wolf Lammen, 15-Jun-2019.)
(∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Theoremsb9i 2425 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Jun-2019.)
(∀𝑥[𝑥 / 𝑦]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)

Theoremax12vALT 2426* Alternate proof of ax12v2 2047, shorter, but depending on more axioms. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsb6 2427* Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. The implication "to the left" is sb2 2350 and does not require any dv condition. Theorem sb6f 2383 replaces the dv condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.)
([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb5 2428* Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. The implication "to the right" is sb1 1881 and does not require any dv condition. Theorem sb5f 2384 replaces the dv condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.)
([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Theoremequsb3lem 2429* Lemma for equsb3 2430. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)

Theoremequsb3 2430* Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.) Remove dependency on ax-11 2032. (Revised by Wolf Lammen, 21-Sep-2018.)
([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)

Theoremequsb3ALT 2431* Alternate proof of equsb3 2430, shorter but requiring ax-11 2032. (Contributed by Raph Levien and FL, 4-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)

Theoremelsb3 2432* Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑦𝑧𝑥𝑧)

Theoremelsb4 2433* Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑧𝑦𝑧𝑥)

Theoremhbs1 2434* The setvar 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by NM, 26-May-1993.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theoremnfs1v 2435* The setvar 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥[𝑦 / 𝑥]𝜑

Theoremsbhb 2436* Two ways of expressing "𝑥 is (effectively) not free in 𝜑." (Contributed by NM, 29-May-2009.)
((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))

Theoremsbnf2 2437* Two ways of expressing "𝑥 is (effectively) not free in 𝜑." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Sep-2018.)
(Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))

Theoremnfsb 2438* If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑧𝜑       𝑧[𝑦 / 𝑥]𝜑

Theoremhbsb 2439* If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by NM, 12-Aug-1993.)
(𝜑 → ∀𝑧𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Theoremnfsbd 2440* Deduction version of nfsb 2438. (Contributed by NM, 15-Feb-2013.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑧𝜓)       (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)

Theorem2sb5 2441* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))

Theorem2sb6 2442* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))

Theoremsbcom2 2443* Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 24-Sep-2018.)
([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Theoremsbcom4 2444* Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2445 but may still be useful. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.)
([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Theorempm11.07 2445 Axiom *11.07 in [WhiteheadRussell] p. 159. The original reads: *11.07 "Whatever possible argument 𝑥 may be, 𝜑(𝑥, 𝑦) is true whatever possible argument 𝑦 may be" implies the corresponding statement with 𝑥 and 𝑦 interchanged except in "𝜑(𝑥, 𝑦)". Under our formalism this appears to correspond to idi 2 and not to sbcom4 2444 as earlier thought. See https://groups.google.com/d/msg/metamath/iS0fOvSemC8/M1zTH8wxCAAJ. (Contributed by BJ, 16-Sep-2018.) (New usage is discouraged.)
𝜑       𝜑

Theoremsb6a 2446* Equivalence for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Sep-2018.)
([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑))

Theorem2ax6elem 2447 We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. This theorem merges two ax6e 2248 instances 𝑧𝑧 = 𝑥 and 𝑤𝑤 = 𝑦 into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd 38594. (Contributed by Wolf Lammen, 27-Sep-2018.)
(¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))

Theorem2ax6e 2448* We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2447 with a distinct variable constraint. (Contributed by Wolf Lammen, 28-Sep-2018.)
𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)

Theorem2sb5rf 2449* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove distinct variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.)
𝑧𝜑    &   𝑤𝜑       (𝜑 ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Theorem2sb6rf 2450* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.)
𝑧𝜑    &   𝑤𝜑       (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Theoremsb7f 2451* This version of dfsb7 2453 does not require that 𝜑 and 𝑧 be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 1837 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1879 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑧𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))

Theoremsb7h 2452* This version of dfsb7 2453 does not require that 𝜑 and 𝑧 be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 1837 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1879 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → ∀𝑧𝜑)       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))

Theoremdfsb7 2453* An alternate definition of proper substitution df-sb 1879. By introducing a dummy variable 𝑧 in the definiens, we are able to eliminate any distinct variable restrictions among the variables 𝑥, 𝑦, and 𝜑 of the definiendum. No distinct variable conflicts arise because 𝑧 effectively insulates 𝑥 from 𝑦. To achieve this, we use a chain of two substitutions in the form of sb5 2428, first 𝑧 for 𝑥 then 𝑦 for 𝑧. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2607. Theorem sb7h 2452 provides a version where 𝜑 and 𝑧 don't have to be distinct. (Contributed by NM, 28-Jan-2004.)
([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))

Theoremsb10f 2454* Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑥𝜑       ([𝑦 / 𝑧]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑))

Theoremsbid2v 2455* An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)
([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)

Theoremsbelx 2456* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
(𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑))

Theoremsbel2x 2457* Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.)
(𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))

Theoremsbal1 2458* A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑦 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 15-May-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2018.)
(¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Theoremsbal2 2459* Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 3-Oct-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Theoremsbal 2460* Move universal quantifier in and out of substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.)
([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)

Theoremsbex 2461* Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)
([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)

Theoremsbalv 2462* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)

Theoremsbco4lem 2463* Lemma for sbco4 2464. It replaces the temporary variable 𝑣 with another temporary variable 𝑤. (Contributed by Jim Kingdon, 26-Sep-2018.)
([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Theoremsbco4 2464* Two ways of exchanging two variables. Both sides of the biconditional exchange 𝑥 and 𝑦, either via two temporary variables 𝑢 and 𝑣, or a single temporary 𝑤. (Contributed by Jim Kingdon, 25-Sep-2018.)
([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Theorem2sb8e 2465* An equivalent expression for double existence. (Contributed by Wolf Lammen, 2-Nov-2019.)
(∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Theoremexsb 2466* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
(∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))

Theorem2exsb 2467* An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 30-Sep-2018.)
(∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))

1.6  Existential uniqueness

Syntaxweu 2468 Extend wff definition to include existential uniqueness ("there exists a unique 𝑥 such that 𝜑").
wff ∃!𝑥𝜑

Syntaxwmo 2469 Extend wff definition to include uniqueness ("there exists at most one 𝑥 such that 𝜑").
wff ∃*𝑥𝜑

Theoremeujust 2470* A soundness justification theorem for df-eu 2472, showing that the definition is equivalent to itself with its dummy variable renamed. Note that 𝑦 and 𝑧 needn't be distinct variables. See eujustALT 2471 for a proof that provides an example of how it can be achieved through the use of dvelim 2335. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))

TheoremeujustALT 2471* Alternate proof of eujust 2470 illustrating the use of dvelim 2335. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))

Definitiondf-eu 2472* Define existential uniqueness, i.e. "there exists exactly one 𝑥 such that 𝜑." Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2508, eu2 2507, eu3v 2496, and eu5 2494 (which in some cases we show with a hypothesis 𝜑 → ∀𝑦𝜑 in place of a distinct variable condition on 𝑦 and 𝜑). Double uniqueness is tricky: ∃!𝑥∃!𝑦𝜑 does not mean "exactly one 𝑥 and one 𝑦 " (see 2eu4 2554). (Contributed by NM, 12-Aug-1993.)
(∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Definitiondf-mo 2473 Define "there exists at most one 𝑥 such that 𝜑." Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 2505. For other possible definitions see mo2 2477 and mo4 2515. (Contributed by NM, 8-Mar-1995.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

Theoremeuequ1 2474* Equality has existential uniqueness. Special case of eueq1 3373 proved using only predicate calculus. The proof needs 𝑦 = 𝑧 be free of 𝑥. This is ensured by having 𝑥 and 𝑦 be distinct. Alternatively, a distinctor ¬ ∀𝑥𝑥 = 𝑦 could have been used instead. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.)
∃!𝑥 𝑥 = 𝑦

Theoremmo2v 2475* Alternate definition of "at most one." Unlike mo2 2477, which is slightly more general, it does not depend on ax-11 2032 and ax-13 2244, whence it is preferable within predicate logic. Elsewhere, most theorems depend on these axioms anyway, so this advantage is no longer important. (Contributed by Wolf Lammen, 27-May-2019.) (Proof shortened by Wolf Lammen, 10-Nov-2019.)
(∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremeuf 2476* A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremmo2 2477* Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.) Restrict dummy variable z. (Revised by Wolf Lammen, 28-May-2019.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremnfeu1 2478 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥∃!𝑥𝜑

Theoremnfmo1 2479 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥∃*𝑥𝜑

Theoremnfeud2 2480 Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Theoremnfmod2 2481 Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Theoremnfeud 2482 Deduction version of nfeu 2484. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Theoremnfmod 2483 Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Theoremnfeu 2484 Bound-variable hypothesis builder for uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝜑       𝑥∃!𝑦𝜑

Theoremnfmo 2485 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
𝑥𝜑       𝑥∃*𝑦𝜑

Theoremeubid 2486 Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Theoremmobid 2487 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Theoremeubidv 2488* Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Theoremmobidv 2489* Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Theoremeubii 2490 Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
(𝜑𝜓)       (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)

Theoremmobii 2491 Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)
(𝜓𝜒)       (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Theoremeuex 2492 Existential uniqueness implies existence. For a shorter proof using more axioms, see euexALT 2509. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2018.)
(∃!𝑥𝜑 → ∃𝑥𝜑)

Theoremexmo 2493 Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.)
(∃𝑥𝜑 ∨ ∃*𝑥𝜑)

Theoremeu5 2494 Uniqueness in terms of "at most one." Revised to reduce dependencies on axioms. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 25-May-2019.)
(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))

Theoremexmoeu2 2495 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)
(∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))

Theoremeu3v 2496* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Add a distinct variable condition on 𝜑. (Revised by Wolf Lammen, 29-May-2019.)
(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Theoremeumo 2497 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.)
(∃!𝑥𝜑 → ∃*𝑥𝜑)

Theoremeumoi 2498 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
∃!𝑥𝜑       ∃*𝑥𝜑

Theoremmoabs 2499 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))

Theoremexmoeu 2500 Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.)
(∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))

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